ABJM Wilson Loops in Arbitrary Representations

We study vacuum expectation values (VEVs) of circular half BPS Wilson loops in arbitrary representations in ABJM theory. We find that those in hook representations are reduced to elementary integrations thanks to the Fermi gas formalism, which are accessible from the numerical studies similar to the partition function in the previous studies. For non-hook representations, we show that the VEVs in the grand canonical formalism can be exactly expressed as determinants of those in the hook representations. Using these facts, we can study the instanton effects of the VEVs in various representations. Our results are consistent with the worldsheet instanton effects studied from the topological string and a prescription to include the membrane instanton effects by shifting the chemical potential, which has been successful for the partition function.


Introduction
Recently, there has been much progress in understanding membranes in M-theory. It was proposed in [1] that the low energy effective theory on the N multiple M2-branes on the geometry C 4 /Z k is described by the 3-dimensional N = 6 supersymmetric generalization of the Chern-Simons matter theory with gauge group U (N ) k × U (N ) −k commonly referred as ABJM theory. Furthermore it has been shown by using the localization technique [2] that a class of supersymmetric observables in the ABJM theory on S 3 are described by so-called ABJM matrix model [3][4][5][6].
The partition function Z(N ) is the first fundamental quantity to be studied. After the rather standard matrix model analysis in [7][8][9], there appeared a seminal paper, which rewrites the ABJM partition function into the partition function of an ideal Fermi gas system [10] (see also [11][12][13]). One of the advantages in this Fermi gas formalism is that instead of the stringy 't Hooft expansion, we can access to the M-theory region directly by taking large N limit with k fixed. As is usual in the statistical system, instead of the partition function, it is convenient to define the grand partition function by introducing the fugacity z = e µ with the chemical potential µ. Subsequently in [14][15][16][17][18][19][20], the partition function of the ABJM theory was studied extensively from this grand partition function of the Fermi gas system. Finally, it turned out that the grand potential J(µ) = log Ξ(z) can be separated into the perturbative, worldsheet instanton [21], membrane instanton [8,22] and bound state part. The worldsheet instanton part is determined directly from the topological string result [17]. The membrane instanton part is also related to the refined topological string [20]. As found in [19], the contributions from all of the bound states can be incorporated to the worldsheet instanton effects by shifting the chemical potential µ to an "effective" chemical potential µ eff , which is described by the sum of µ and a part of the pure membrane instanton effects.
Here we proceed to study the second fundamental quantity, namely, the vacuum expectation value (VEV) of the circular half BPS Wilson loop 1 firstly introduced in [23,24]. The half BPS Wilson loops have nice counterparts in the open topological string, as was pointed out in [7,24]. This is one of our motivation that we focus on them here. The half BPS Wilson loops are classified by representations R of the supergroup U (N |N ), which includes the gauge group U (N ) × U (N ) as the bosonic subgroup. By using the localization method [3][4][5][6], the unnormalized VEV of the Wilson loop W R in the representation R is written as where g s = 2πi k is the coupling constant, and Str R is the U (N |N ) character in the representation R. A prescription to obtain Str R is summarized as follows. First, a representation of the supergroup U (N |N ) is characterized by the super Young diagram, which has the same form as the usual Young diagram of the bosonic group U (∞) (for example, see [25]). Then, the supertrace Str R U of the supergroup U (N |N ) is found if we formally replace a 2 a 1 (a) Partition notation (b) Frobenius notation Figure 1. (a) The partition notation [λ 1 λ 2 λ 3 · · · ] with its transpose [λ ′ 1 λ ′ 2 λ ′ 3 · · · ] and (b) the Frobenius notation (a 1 a 2 · · · a r |l 1 l 2 · · · l r ) for the same Young diagram. Here r = max{s|λ s − s ≥ 0} = max{s|λ ′ s − s ≥ 0} is the number of diagonal boxes, and a p , l q denote the horizontal and vertical distances from each diagonal box, respectively, given by a p = λ p − p, l q = λ ′ q − q. In the above case, the Young diagram is [λ 1 λ 2 λ 3 λ 4 ] = [5,3,3,2] in the partition notation with its [4,4,3,1,1], while it is (a 1 a 2 a 3 |l 1 l 2 l 3 ) = (4, 1, 0|3, 2, 0) in the Frobenius notation.
the power sum tr U n in tr R U of U (∞) by Str U n . Note that U appearing in Str R U is a 2N × 2N matrix defined by (1.2).
The computation of the VEVs using the Fermi gas formalism was initiated in [26], where the inserted observables are restricted to the operators with winding number n, Str U n . Very recently, it was proposed in [27] that it is possible to study the perturbative part and the worldsheet instanton part using the topological strings. This subject keeps on attracting various studies. 2 In this paper, we present a Fermi gas formalism for the VEVs in arbitrary representations, suitable for numerical study, and study these non-perturbative effects. As in the partition function, besides the worldsheet instanton contribution, we also find the contribution coming from the membrane instanton, which is difficult to be known from the topological string theory. In the following of this introduction, we would like to explain our results in more details. Just as in the partition function, it is useful to consider the VEV in the grand canonical ensemble defined by Note that once we know W R GC , the VEVs in the canonical ensemble is easily recovered.
First, we find a formula for the VEV of the Wilson loop in the hook representation 3 2 See, for example, [28] for perturbative studies of the Wilson loop VEVs, [29,30] for the holographic studies, [30][31][32] for generalizations of contour and [33] for more general Chern-Simons matter theory. 3 Throughout this paper, we use the Frobenius notation to express representations of U (N |N ) illustrated in Figure 1.

R = (a|l) in terms of a certain convolution of integrations
Here ρ 1 is the density operator of the Fermi-gas system defined later in (2.9), and the states a| and |l in the coordinate basis are given by (3.9). The expression (1.4) is accessible from the numerical studies with very high precision. Second, we extend our analysis to general representation R = (a 1 a 2 · · · a r |l 1 l 2 · · · l r ). In operator level, the Wilson loop is simply given by the determinant of those in the hook representations, known as the Giambelli formula, 4 W (a 1 a 2 ···ar|l 1 l 2 ···lr) (e µ , e ν ) = det p,q W (ap|lq) (e µ , e ν ). (1.5) In this paper, we find that the VEVs in the grand canonical ensemble exactly satisfy the same type of the formula, Hence the VEVs of the half BPS Wilson loops in general representations can be computed only from those in hook representations. We emphasize that this result is very unexpected and non-trivial. In the mathematical context, a normalized linear functional O of symmetric functions O satisfying the above property is called Giambelli compatible (see e.g. [34]). Let us further call a linear functional being factorizable if it satisfies the property Note that the factorizability implies the Giambelli compability. In this terminology, we show that the grand canonical VEV of the half BPS Wilson loop is Giambelli compatible but not factorizable. We also find that its perturbative part is factorizable (see (5.10)). Note that the factorization of the grand canonical VEV also implies that of the canonical VEV in the large N limit, which is natural from the physical viewpoint. The factorization property, however, is generically broken by the instanton contributions. Nevertheless, the Giambelli compatibility is still preserved after the instanton effects are taken into account. Finally, using our results (1.4) and (1.6), we also study the structure of the instanton corrections to the VEVs in various representations by the numerical studies. The VEVs, in general, receive the following corrections, where W is the worldsheet instanton correction, and W GC(others) R (µ, k) consists of the pure membrane instanton correction and the contribution from the bound states. We have found that our numerical results match with the topological string prediction of the perturbative part and the worldsheet instanton part with the chemical potential shifted from µ to µ eff to incorporate the contribution from the membrane instantons and the bound states: exactly the same as in the partition function. Here the "effective" chemical potential µ eff was introduced in [19] in order to explain the bound state contribution in the grand potential, where a ℓ (k) are the functions appearing in the membrane instanton correction in the grand potential. The forms of a ℓ (k) are exactly computed by the refined topological string on local P 1 ×P 1 [20]. We should stress that the perturbative part and the worldsheet instanton part in (1.8) are computed from the open topological string on local P 1 × P 1 as we will see in section 5. Thus our result states that once we determine the topological string free energy on this background, we can exactly find the VEVs of the half BPS Wilson loops in general representations in the ABJM theory. The organization of this paper is as follows. In the next section we present a general framework to study the VEVs of the BPS Wilson loops and apply it to the half BPS case in the hook representation in section 3. Since it is difficult to apply this formalism directly to the half BPS Wilson loops in the non-hook representation, we shall present an alternative method in section 4, which works only for the half BPS Wilson loop. After reviewing the results from the topological strings in section 5, we summarize our numerical study in section 6. Finally we conclude in section 7.

BPS Wilson loops in general representations
Here we present methods to study the VEV of the Wilson loop in the ABJM theory using the Fermi gas formalism. We shall first present a framework to study general 1/6 BPS Wilson loop constructed in [35], which includes the half BPS Wilson loop as a special case.

Partition function
For this purpose let us first review the derivation of the Fermi gas formalism for the partition function [10] carefully because our Wilson loop insertion is based heavily on it. The starting point is the partition function of the ABJM matrix model [3][4][5][6]: By using the Cauchy identity and performing a Fourier transformation, the partition function (2.1) is rewritten into where we rescale the integration variables as µ i = x i k , ν j = y j k and = 2πk. After performing the Gaussian integral over x and y by completing the square in the exponent and noting the cancellation of the p 2 and q 2 terms, the partition function becomes If we further integrate over p in (2.4), then we find (2.5) Since the partition function Z(N ) has the form of an ideal Fermi gas system as it is easier to consider the grand canonical partition function (1.1) by introducing the fugacity z = e µ . One can show that the grand partition function is expressed as a Fredholm determinant, Ξ(z) = Det(1 + zρ 1 ), (2.8) where the determinant Det is taken over the whole Hilbert space of the Fermi gas system. In the operator formalism, the density matrix ρ 1 is given by where q and p satisfies the canonical commutation relation [q, p] = i with = 2πk. We adopt this notation in what follows.

Operator insertion
General 1/6 BPS Wilson loops in the ABJM theory are generated by the following type of operator [3]: where f (x) and g(x) are functions of x. In this section we translate the insertion of this operator into the one of a certain quantum mechanical operator expressed by (q, p). As a warm up, let us first consider the operator insertion into the partition function (2.2). After completing the square in integrating over x M and combining with the contribution from integrating y M as in the computation of the partition function, we find an extra contribution into the exponent: Performing the integration over p M , the unnormalized VEV is finally given by (2.13) In the language of quantum mechanical operators, the second line can be interpreted as the matrix element Therefore we conclude that the insertion of the operator e nµ M amounts to the insertion of the operator W n to the right of P , where W is defined by Similarly, we find that the insertion of the operator e nν M amounts to insertion of the same operator W n to the left of P . This can be seen by repeating the square completion in the exponent with an extra factor and computing of the matrix element Note that this interpretation is factor-wise. Namely, not only other additive terms in the insertion do not affect this interpretation, but this interpretation is valid even if this operator is multiplied by other operators. We can also see that the simultaneous insertion at the same position M , namely, e mµ M +nν M also works well. Therefore we can summarize the computation rule as follows. For the case of the partition function, we finally end up with the summation over the conjugacy classes and the study of For the case of Wilson loop, we insert W into various slots between Q and P in this trace. The insertion pattern depends on the representation, but since we are considering the gauge invariant operator, we have to take a trace, namely, sum over all the insertion slots. Hence our formula can be summarized as where O GC denotes the expectation value of the operator O in the grand canonical ensemble (1.3). Once the grand canonical VEV is understood, one can easily return to the canonical VEV via (2.20) Alternatively, we can show the relation (2.19) using the operator formalism as follows. The expectation value of i f (e µ i )g(e ν i ) at fixed N is given by (2.21) where ρ denotes the density matrix in the presence of operator insertion This can be written as an operator equation where we have used Therefore, up to a similarity transformation the density matrix in (2.24) becomes

Half BPS Wilson loops I: hook representations
In the previous section, we have presented a general framework to study the VEVs of the general 1/6 BPS Wilson loop in the Fermi gas formalism. Especially we have reduced the problem into computing the trace with alternating operators Q and P and various W -insertions. This quantity, however, is still difficult to compute, at least, numerically with high precision. Here we would like to see what kind of simplification will occur if we restrict ourselves to the half BPS Wilson loops.

Representations of the superalgebra
The half BPS Wilson loop is classified by the representation of U (N |N ) [23,24]. In this subsection we review representations of the supergroup U (N |N ). For this purpose, it is convenient to consider representations of U (∞). A simple prescription to derive the character of U (N |N ) is to formally replace tr U n in the character tr R U of U (∞) by Str U n : Note that the character tr R U is given by the Schur function associated with the Young diagram R. The supertrace Str R U can be expressed by a combination of characters of two bosonic subgroups U (N ) of U (N |N ). For example, in the case of the 2nd anti-symmetric representation (0|1), the superalgebraic generalization turns out to be where U µ and U ν are the bosonic parts of U (see (1.2)). Below, we often denote the supertrace Str R U by W R (e µ , e ν ), and use the abbreviation W R = W R (e µ , e ν ) as long as there is no risk of confusion.

Beyond winding Wilson loops
The Wilson loop with the winding number n was studied extensively in [26]. By revisiting this in our formalism, we will obtain a hint to study the more general representations as in the following. In our formalism, applying the rule in (2.19) with the choice, , (3.4) and picking up the linear term in t, the grand canonical VEV of Str U n is given by with R(z) defined by One can easily see that the operator appearing on the right-hand-side is expanded as Note that the operator appearing in the right-hand-side of (3.7) has the factorized form where the coordinate q representations of |n and m| are defined by As a formal operator relation, (3.8) is also written as Thus we finally obtain the grand canonical VEV of the winding Wilson loop (3.5) as Comparing with the relation between the winding Wilson loop Str U n and the Wilson loop W (a|l) in the hook representation it is tantalizing to expect the relation which is true as we will see in the next subsection. More generally, the computation of the VEVs of the half BPS operators reduces to picking up a certain function f (W ) and computing the Fredholm determinant of the corresponding density matrix ρ f Rewriting the density matrix in the above expression as and recalling (3.7), one can see that the grand canonical VEV of the half BPS Wilson loops can always be written as a sum of the factorized functions.

Single-hook representations
For the half BPS Wilson loop in a single-hook representation (a|l), the generating function is given by [36] 1 When plugging into our formula (3.14), we find that the corresponding density matrix factorizes as s a t l |l a|.
Therefore, the grand canonical VEV of (3.16) becomes Finally, the grand canonical VEV of W (a|l) is found to be (3.13) which is accessible from the numerical studies similar to the partition function in the previous studies [15][16][17]19].

Half BPS Wilson loops II: general representations
In the previous sections, we have presented a method to compute the supersymmetric Wilson loops and shown that especially for the half BPS Wilson loop in the hook representation, there is a factorization, which at least simplifies the numerical study. The above analysis for the hook representation is, however, difficult to be extended to a general non-hook case. Here we shall present a completely different analysis which is effective for studying the non-hook representations from the hook representations but only suitable for the half BPS Wilson loop.

Non-hook representations
After understanding the VEV in the hook representation in the previous section, we can go beyond the hook representation step by step. Namely, we can substitute various functions for f (W ) and subtract the known hook part. For example, if we plug f (W ) = e tW , which corresponds to the generating function of (StrU ) n , and compare O(t 4 ) terms, then we find More generally, it is easy to imagine the expression in (1.6). By changing the function for f (W ), we will encounter various relations supporting this conjecture. However, it is difficult to prove it directly using this formulation.
Therefore, we would like to study W (a 1 a 2 ···ar|l 1 l 2 ···lr) (e µ , e ν ) = det p,q W (ap|lq) (e µ , e ν ) . Instead of computing it directly, here let us consider and picking up the coefficient of the highest t r term. The reason we want to consider W (N ) is because this is a generalization of the Cauchy determinant The proof of this formula for r = 1 is simply reduced to a more general formula in [37]. 6 The proof for r > 1 is reduced to the case of r = 1 by the formula Then the quantity we want to compute becomes 5 We are grateful to Sho Matsumoto for his collaborative contribution in sharing his idea of proof and the references with us in this subsection. 6 The formula of [37] for the r = 1 case is written as where M −1 is the inverse of Cauchy matrix Mij = 1/(xi + yj). One can show that the generating function of (4.5) reproduces (3.16). (4.8) Using the formula (4.4), we can rewrite this as (4.10) Since the VEV can be interpreted as the partition function of the ideal Fermi gas system just as the partition function (2.6), it is natural to introduce the generating function as where Det is defined through the trace over the indices ν with the measure in (4.7). Therefore, if we define then we find where the multiplication among variables in the boldface character are understood as matrix multiplication with indices µ, ν and measures in (4.7). Note that the square root √ Q should be regarded as a formal notation. We can express the integrations without it. The reason we introduce it is because of the relation to the previous quantities as we shall see below. Here, in the last equation we have used the formula which is the same as This holds for both the hook and the non-hook cases. Now using this result (4.15) we can reduce the proof of (1.6) to the result of (1.4) given in the previous section or we can prove (1.4) independently. Let us first consider to reduce to the previous result. If we pick up the constant term by taking the limit t → 0, we find Comparing with the expression for the partition function (2.8), we find Also, if we apply the above results to the single-hook case, we find Again comparing with the expression for the hook representation we have, we find Instead of our comparison with the known results, the argument here also suggests that if we restrict ourselves to the half BPS Wilson loop, we can have an alternative derivation for the hook case if we evaluate carefully Det(1 + zρ 1 ) and a p |(1 + zρ 1 ) −1 |l q . The computation of Det(1 + zρ 1 ) is exactly what we did around (2.4). Also, the computation of a|(1 + zρ 1 ) −1 |l becomes dx e ix 2 /(2 ) · · · dy e −iy 2 /(2 ) e 2π(a+ 1 2 )x/ e −iqx(x−y ′ )/ 2 cosh qx 2 · · · e −iqy(x ′ −y)/ 2 cosh qy 2 e 2π(l+ 1 2 )y/ , (4.21) In completing the square for x and y we find

(4.22)
Note that q 2 terms cancel with the square completion from the neighboring terms. Hence, we are left with 1 2k ((a + 1/2)q x + 2πia(a + 1) + (l + 1/2)q y − 2πil(l + 1)). (4.23) This is nothing but the exponent we found in (3.8) with (3.9). We note in passing that the above computation can be done also in the operator formalism.

Fermionic representation
Our general expression (4.20) of the Wilson loop VEV suggests that there is an underlying fermionic structure. This is expected from the fermionic nature of D-branes in topological string theory [38]. Introducing the fermions with the standard anti-commutation relation such that the vacuum is annihilated by the positive modes as we define the state |V as k . Using the q-binomial formula, one can show that the alternating sum of (4.29) reproduces the perturbative part of winding Wilson loop [26] a+l=n−1 (4.30)

Relation to open topological strings
In this section, we see a relation between the VEVs of the half BPS Wilson loops and the open topological string amplitudes. As is well-known, the ABJM matrix model is related to the L(2, 1) lens space matrix model by analytic continuation [7,24] (see also [39,40]). This lens space matrix model is also related to the topological string on local P 1 × P 1 through the large N duality [41]. In fact, the perturbative and the worldsheet instanton parts in the ABJM partition function can be captured by the result of the closed topological string on local P 1 × P 1 . Similarly, the VEVs of the half BPS Wilson loops are described by the open topological string. Here we are interested in the VEVs in the grand canonical ensemble, which corresponds to the so-called large radius frame on the topological string side. The open topological string in this frame was recently studied in detail in [27]. We note that the membrane instanton corrections are difficult to be known from the topological string because these corrections correspond to the non-perturbative effects in the topological string. We will explore the membrane instanton corrections in the next section with the help of the numerical analysis.
First we briefly summarize the result of [27]. The open topological string amplitudes take the following general form [42][43][44], where t is the Kähler moduli of the local Calabi-Yau X, and V is the open string moduli. For the ABJM theory, we are interested in local P 1 × P 1 . The string coupling in the topological string is related to the Chern-Simons level, There are two Kähler moduli, which are identified as the chemical potential µ dual to the original rank N , Similarly, the open string moduli V is also identified with the dual variable for the Wilson loop insertion U . Then, we can relate the perturbative part and worldsheet instanton part of the grand canonical VEVs in the ABJM theory to the above open topological string amplitudes. The concrete relation is given explicitly by [27], with c n 1 ,n 2 ,... = 1/( j j n j n j !). Note that to write down the relation we have to plug a new parameter into (5.1).
Therefore we immediately find (5.9) and the factorization property (Str U ) n 1 (Str U 2 ) n 2 · · · GC(pert) = ( Str U GC(pert) ) n 1 ( Str U 2 GC(pert) ) n 2 · · · . (5.10) Note that this factorization property does not hold if the instanton effect is taken into account. One can check that these results reproduce (4.29) for the hook representations. From the factorization property (5.10), one finds that the perturbative part of the half BPS Wilson loop in the representation R scales as (5.11) where n is the number of boxes of Young diagram R and we have dropped the prefactor independent of µ. Coming back to the VEV in the canonical ensemble via (2.20), we find that the perturbative part of the half BPS Wilson loop in arbitrary representation gives the following Airy function behavior where the proportional coefficient depends only on k. From this expression, we can also find the large N limit as (N → ∞), (5.13) where λ = N/k is the 't Hooft coupling. Note that this exponent is the same as n times of an classical string action on the gravity side [23].

Worldsheet instantons
Let us consider the worldsheet instanton corrections. We first denote the general open string amplitude by After specifying β = (d 1 , d 2 ) and take the "diagonal" sum for the open GV invariants Thus we find from (5.14), for example, where the relation between V and V is given by (5.5). Tables 1 and 2 of [27], we obtain the worldsheet instanton corrections up to order Q 5 ,

Using the explicit values of the open GV invariants listed in
As discussed in [45], the g = 0 terms of Str U GC(pert+WS) are given by the factor (Q/z) 1 2 representing the worldsheet instanton corrections to the disk amplitude. Here z and Q are related by the mirror map of local P 1 × P 1 along the diagonal slice z 1 = z 2 = z Inverting this relation, the worldsheet instanton corrections to the disk amplitude are found to be which reproduce the invariants n g=0,d,(1) listed in [27]. Interstingly, we find from [46] that the VEV of the Wilson loop with widing n is generically written as the following form, where N g em,d are integers, which are related to the open GV invariants n g,d,ℓ . Instead of the topological string consideration, we can also fix such integers by comparing the matrix model results [7,24,26] in the 't Hooft limit because the genus expansion in this limit captures all the worldsheet instanton corrections. The similar comparison on the worldsheet instanton corrections to the grand potential has been done in [17]. In this way, we have fixed the values of N g em,d in the very first few cases. The result is summarized in Table 1. For n = 1, 2, one can check that (5.22) with Table 1 indeed reproduce (5.19).
In the next section, we will confirm that these worldsheet instanton corrections are indeed consistent with our numerical results.

Numerical study and membrane instantons
In this section, we numerically evaluate the VEVs of the half BPS Wilson loops in hook representations by using the formulation presented in sections 3 and 4. The main motivation of this analysis is to explore the membrane instanton effects, which are very hard to be described in the topological string theory. The similar analysis has been already done for the grand partition function in [17,19]. We compute the VEVs in various hook representations for some values of k. Here we propose that the membrane instanton corrections are completely encoded by the replacement µ → µ eff in the perturbative part and  The effective chemical potential µ eff is explicitly given [19] for even k = 2n as and conjectured for odd k as Below, we will check the proposal (1.8) by the numerical study.

A procedure
Let us consider the VEV for the half BPS Wilson loops in the hook representation (a|l). The VEV is given by (1.4), Let us first note that, the complex phase dependence only come from a|x and y|l , which is trivially given in (3.9), Hence we define a real function W (a|l) with its series expansion W (m) (a|l) as where W (m) (a|l) is given by Of course, the VEVs for (a|l) and (l|a) should be complex conjugate to each other, therefore we immediately find Our task is to evaluate the integral (6.6). This can be done as follows. Let us introduce the function by One easily finds that this function satisfies the recurrence relation (6.9) with the initial condition (6.10) Once the function φ (m) l (x) is known, the integral (6.6) is easily evaluated as We notice that the integral equation (6.9) is essentially the same as that appearing in [15][16][17]. One can solve it for any k at least numerically. Practically, we solve the integral equation up to a certain value m = m max , and make an approximation for W (a|l) (6.5) as Though this approximation is originally valid only for the small µ regime, we extrapolate the profile to a reasonably large µ regime and fit the expansion coefficients of W (a|l) by exact values at this regime. This is the same strategy as that in [17]. Before closing this subsection, we will briefly comment on the convergence of integral. In (6.10) and (6.11), there appear the exponential factors that diverge in large x limit. Due to these factors, the integral (6.11) converges only if k > 2(a + l + 1) = 2|R hook |, (6.13) where |R hook | is the size of Young diagram corresponding to the hook representation R hook . Therefore, the grand canonical VEVs, and correspondingly the canonical VEVs W R N , are well-defined only for such values of k, though it is much harder to see it directly in W R N due to the complex phases in the original expression (1.2). Such a behavior has also been found for the multiple winding Wilson loop in [26].

Fundamental representation
The simplest representation is the fundamental representation (0|0), 14) We would like to evaluate W (0|0) numerically for some values of k. By solving the integral equation, we have performed the numerical computation, and find the non-perturbative corrections to W (0|0) for k = 3, 4, 6, 8, 12. The results are as follows: , , , To obtain the numerical results, we have chosen m max to be the best value of the numerical fitting at each instanton number. This best value decreases as the instanton number increases because of the exponential suppression of the corrections and the numerical errors. Anyway, for all the above cases, we have chosen m max ∼ 10, and the numerical results of the coefficients match to these exact values (6.15) in about 5-digit accuracy. Especially, if we make a wrong guess at a certain instanton order, the next order of instanton would grow exponentially, which could make our fitting totally impossible. Note that for the above values of k the coefficients of (6.16) become particularly simple (especially rational) because the root of unity e 2πi/k takes the simple values. This is why we choose the above values of k for our fitting problem. Let us compare these results with the theoretical prediction. The worldsheet instanton corrections of W (0|0) GC = Str U GC are given by (5.19). As mentioned before, we propose that the membrane instanton correction can be incorporated by the replacement µ → µ eff in the worldsheet instanton correction. Thus our conjecture, including the membrane instanton effects, is where Q eff = −e − 4µ eff k . For the comparison, we need to rewrite it in terms of Q = −e − 4µ k . Using the relations (6.1) and (6.2) between µ and µ eff , we find Plugging these into (6.17), one can check that the corrections exactly agree with the numerical ones (6.15) up to Q 5 . We emphasize that only the worldsheet instanton correction does not explain the numerical results (6.15). We need to replace µ by µ eff to reproduce them. This is due to the membrane instanton effects.

Young diagrams with two boxes
There are two representations with two-box Young diagrams: We have the relation W (1|0) = W (0|1) . By the similar computation to the fundamental representation, we find , , Using the relation (6.18), one finds that the corrections again agree with the numerical ones (6.20) up to Q 5 .

Young diagrams with three boxes
For the three-box Young diagrams, there are two non-trivial real functions, with the constraint W (2|0) = W (0|2) . From the numerical analysis, we find , and , .  One can check that this reproduces the above results for k = 8, 12 up to Q 3 .

Conclusion
In this paper we have proposed the Fermi gas formalism for the VEVs of the half BPS Wilson loops in arbitrary representations. For the case of the hook representations, we present the formula in terms of the convolution of integrations. For the case of the non-hook representations, we reduce the computation to the hook case by a determinant formula similar to the Giambelli formula for the Schur polynomial. After working out these expressions for the VEVs, we also present a numerical study. We find that besides the worldsheet instanton corrections we also have the membrane instanton corrections which can be incorporated by shifting the chemical potential µ into µ eff as we did in studying the bound states in the ABJM partition function. We conclude our paper by listing several discussions on the further directions. Based on the numerical results, we conclude that the membrane instanton correction is completely encoded in the perturbative and the worldsheet instanton parts by replacing µ by µ eff . Let us recall that in the partition function, there is also a pure membrane instanton correction, as well as the bound states of the worldsheet instantons and the membrane instantons. This pure membrane instanton correction is directly related to the non-perturbative effect in the closed topological string [20] (see also [47]). Our Wilson loop result (1.8) implies that there seem to be no pure membrane instanton corrections in the open topological string on "diagonal" local P 1 × P 1 . It would be interesting to confirm this in the topological string framework.
Most of our analysis here focus on the half BPS Wilson loops, which have nice counterparts in the topological string. Our method presented in section 2, however, can be applicable to the 1/6 BPS Wilson loops. The topological string counterparts to such 1/6 BPS Wilson loops are unclear, thus it would be important to reveal the structure of instanton effects in the 1/6 BPS Wilson loops by using our method. It is also interesting to perform Monte Carlo simulation [48] of the 1/6 BPS Wilson loops in low dimensional representations, which has been useful for the partition function [14]. It would also be illuminating to apply our formalism to other observables in the ABJM theory such as the vortex loop [6] and energy-momentum tensor correlator [49], which can be also simplified by the localization method.
In the topological string theory, we have a set of open GV invariants for each representation. In our Fermi gas formalism, we find several non-trivial relations among such invariants. The simplest one is the symmetry of taking the transpose in the Young diagram. For example, disregarding a difference in the trivial phase factor, the VEVs of the half BPS ABJM Wilson loops in the symmetric and anti-symmetric representations are equal with each other. Hence this triviality of the phase factor imposes highly non-trivial relations in the open GV invariants. The origin of this property is unclear on the topological string side at present. Besides, the VEVs in the non-hook representations enjoy the Giambelli property. Technically, the Giambelli property imposes many interesting relations and reduces largely the unknown open GV invariants. Using the Giambelli property, we can show that the number of unknown GV invariants at each order reduces to the number of boxes n, which originally increases with the number of representations, namely, partitions p(n) ∼ e π 2n 3 /(4 √ 3n). It is interesting to clarify what kind of relations the transposition symmetry and the Giambelli compatibility will impose on the open GV invariants. We also ask whether these kinds of relations appear in more general topological string theories or not. Since we have studied only the local P 1 × P 1 topological string, the relations might be accidental properties in this model. If these are common in a class of topological strings, we expect that there are some extra structures, which naturally explain the relations. For example, since the topological recursion of Eynard and Orantin [50] gives relations among all open string invariants, this might explain the relations coming from the transposition symmetry and the Giambelli compatibility.
A natural open question is the physical interpretation of the Giambelli compatibility. It would be nice to understand its meaning from the brane configuration or the gravity analysis. We hope that this would be a clue to understand M-theory.